Problem: Julia has a farm on a rectangular piece of land that is $150$ meters long. This area is divided into two parts: A square area where she lives (whose side is the same as the width of the farm), and the remaining area where she grows artichokes. Every week, Julia spends $\$8$ per square meter on the area where she lives, and earns $\$4$ per square meter from the area where she grows artichokes. This way, she manages to save some money every week. Write an inequality that models the situation. Use $w$ to represent the width of Julia's farm.
Answer: The strategy We know that Julia saves some money each week. In other words, her spending is less than her earnings. So, if $X$ is the amount of money she spends on her home each week and $Y$ is the amount of money she earns from her artichokes each week, then $X<Y$. Now, let's express $X$ and $Y$ in terms of $w$. Expressing the amount of money spent Since the residential area of Julia's farm is a square, and since $w$ is the width of land, the area of the region in which Julia lives is $w^2$ square meters. Each week, Julia spends $\$8$ per square meter on the area in which she lives, and so Julia spends $8w^2$ dollars each week. Expressing the amount of money earned Since Julia's farm is $150$ meters long, and since the residential area uses $w$ meters of that length, the area where Julia grows artichokes is $150-w$ meters long. Its width is of course $w$ meters, and so the total area of the artichoke section of her farm is $w(150-w)$ square meters. Each week Julia earns $\$4$ per square meter from the area in which she grows artichokes, so in total she earns $4w(150-w)$ dollars each week. Putting things together We found that $X=8w^2$ and $Y=4w(150-w)$. Since $X<Y$, we can substitute and find an inequality in terms of $w$ that models the situation. The answer is: $ 8w^2<4w(150-w)$